Partial Differential Equations in Several Complex Variables

This book is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. In the last few decades, significant progress was made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables. After the background material in complex analysis is developed in Chapters 1 through 3, the next three chapters are devoted to the solvability and regularity of the Cauchy-Riemann equations using Hilbert space techniques. The authors provide a systematic study of the Cauchy-Riemann equations and the $\bar\partial$-Neumann problem, including Hórmander's $L^2$ existence progress on the global regularity and irregularity of the $\bar\partial$-Neumann operators. The second part of the book gives a comprehensive study of the tangential Cauchy-Riemann equations, another important class of equations in several complex variables first studied by Lewy. An up-to-date account of the $L^2$ theory for $\bar\partial_b$ operator is given. Explicit integral solution representations are constructed both on the Heisenberg groups and on strictly convex boundaries with estimates in Hölder and $L^2$ spaces. Embeddability of abstract $CR$ structures is discussed in detail here for the first time.

This fairly self-contained book provides a much-needed introductory text to several complex variables and PDEs. It also provides a rich source of information to experts.

Titles in this series are copublished with International Press, Cambridge, MA.

Contents

• Real and complex manifolds
• The Cauchy integral formula and its applications
• Holomorphic extension and pseudoconvexity
• $L^2$ theory for $\overline\partial$ on pseudoconvex domains
• The $\overline\partial$-Neumann problem on strongly pseudoconvex manifolds
• Boundary regularity for $\overline\partial$ on pseudoconvex domains
• Cauchy-Riemann manifolds and the tangential Cauchy-Riemann complex
• Subelliptic estimates for second order differential equations and $\square_b$
• The tangential Cauchy-Riemann complex on pseudoconvex $CR$ manifolds
• Fundamental solutions for $\square_b$ on the Heisenberg group
• Integral representations for $\overline\partial$ and $\overline\partial_b$
• Embeddability of abstract $CR$ structures
• Appendix
• Bibliography
• Table of notation
• Index